Some users suggested that probabilists think of random variables (r.v.'s) as variables or as numbers, rather than functions. This sounds interesting to me, as I have written a number of papers in probability, but hardly ever thought of r.v.'s as variables or numbers.
To me, the usefulness of r.v.'s is mainly in the convenience of notation. It is a bit simpler to write and, I think, to grasp $\mathbb E X$ than $\int_{\mathbb R}x\,\mu(dx)$, where $\mu$ is the distribution of a r.v. $X$.
Similarly, it is simpler to write and to grasp $\mathbb P(X+Y\le s)$ than $\nu(\{(x,y)\in\mathbb R^2\colon x+y\ge s\})$$\nu(\{(x,y)\in\mathbb R^2\colon x+y\le s\})$, where $\nu$ is the distribution of a pair $(X,Y)$ of r.v.'s. (Of course, $\mathbb P(X+Y\le s)$ is a convenient abbreviation for $\mathbb P(\{\omega\in\Omega\colon X(\omega)+Y(\omega)\le s\})$.)
Also, outside of mathematics, r.v.'s are what usually comes first in mathematical modeling. Say, first one models errors of measurements as random variables and then thinks how to model the (joint) distribution of those random variables!